Fermi's golden rule


In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time (so long as the strength of the perturbation is independent of time) and is proportional to the strength of the coupling between the initial and final states of the system (described by the square of the matrix element of the perturbation) as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

Historical background Edit

Although the rule is named after Enrico Fermi, most of the work leading to it is due to Paul Dirac, who twenty years earlier had formulated a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.[1][2] It was given this name because, on account of its importance, Fermi called it "golden rule No. 2".[3]

Most uses of the term Fermi's golden rule are referring to "golden rule No. 2", but Fermi's "golden rule No. 1" is of a similar form and considers the probability of indirect transitions per unit time.[4]

The rate and its derivation Edit

Fermi's golden rule describes a system that begins in an eigenstate   of an unperturbed Hamiltonian H0 and considers the effect of a perturbing Hamiltonian H' applied to the system. If H' is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If H' is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency ω, the transition is into states with energies that differ by ħω from the energy of the initial state.

In both cases, the transition probability per unit of time from the initial state   to a set of final states   is essentially constant. It is given, to first-order approximation, by

where   is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial states, and   is the density of states (number of continuum states divided by   in the infinitesimally small energy interval   to  ) at the energy   of the final states. This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Thus, the probability of finding the system in state   is proportional to  .

The standard way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition.[5][6]

Only the magnitude of the matrix element   enters the Fermi's golden rule. The phase of this matrix element, however, contains separate information about the transition process. It appears in expressions that complement the golden rule in the semiclassical Boltzmann equation approach to electron transport.[9]

While the Golden rule is commonly stated and derived in the terms above, the final state (continuum) wave function is often rather vaguely described, and not normalized correctly (and the normalisation is used in the derivation). The problem is that in order to produce a continuum there can be no spatial confinement (which would necessarily discretise the spectrum), and therefore the continuum wave functions must have infinite extent, and in turn this means that the normalisation   is infinite, not unity. If the interactions depend on the energy of the continuum state, but not any other quantum numbers, it is usual to normalise continuum wave-functions with energy   labelled  , by writing   where   is the Dirac delta function, and effectively a factor of the square-root of the density of states is included into  .[10] In this case, the continuum wave function has dimensions of  , and the Golden Rule is now

where   refers to the continuum state with the same energy as the discrete state  . For example, correctly normalized continuum wave functions for the case of a free electron in the vicinity of a hydrogen atom are available in Bethe and Salpeter.[11]
Normalized Derivation in time-dependent perturbation theory

The following paraphrases the treatment of Cohen-Tannoudji.[10] As before, the total Hamiltonian is the sum of an “original” Hamiltonian H0 and a perturbation:  . We can still expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system, but these now consist of discrete states and continuum states. We assume that the interactions depend on the energy of the continuum state, but not any other quantum numbers. The expansion in the relevant states in the Dirac picture is

where  ,   and   are the energies of states  . The integral is over the continuum  , i.e.   is in the continuum.

Substituting into the time-dependent Schrödinger equation

and premultiplying by   produces
where  , and premultiplying by   produces
We made use of the normalisation  . Integrating the latter and substituting into the former,
It can be seen here that   at time   depends on   at all earlier times  , i.e. it is non-Markovian. We make the Markov approximation, i.e. that it only depends on   at time   (which is less restrictive than the approximation that   used above, and allows the perturbation to be strong)
where   and  . Integrating over  ,
The fraction on the right is a nascent Dirac delta function, meaning it tends to   as   (ignoring its imaginary part which leads to an unimportant energy shift, while the real part produces decay [10]). Finally
which has solutions:  , i.e., the decay of population in the initial discrete state is   where

Applications Edit

Semiconductors Edit

The Fermi golden rule can be used for calculating the transition probability rate for an electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor, and also for when the electron recombines with the hole and emits a photon.[12] Consider a photon of frequency   and wavevector  , where the light dispersion relation is   and   is the index of refraction.

Using the Coulomb gauge where   and  , the vector potential of the EM wave is given by   where the resulting electric field is


For a charged particle in the valence band, the Hamiltonian is

where   is the potential of the crystal. If our particle is an electron ( ) and we consider process involving one photon and first order in  . The resulting Hamiltonian is
where   is the perturbation of the EM wave.

From here on we have transition probability based on time-dependent perturbation theory that

where   is the light polarization vector. From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the bracket.

For the initial and final states in valence and conduction bands respectively, we have   and  , and if the   operator does not act on the spin, the electron stays in the same spin state and hence we can write the wavefunctions as Bloch waves so

where   is the number of unit cells with volume  . Using these wavefunctions and with some more mathematics, and focusing on emission (photoluminescence) rather than absorption, we are led to the transition rate
where   is the transition dipole moment matrix element is qualitatively the expectation value   and in this situation takes the form

Finally, we want to know the total transition rate  . Hence we need to sum over all initial and final states (i.e. an integral of the Brillouin zone in the k-space), and take into account spin degeneracy, which through some mathematics results in


where   is the joint valence-conduction density of states (i.e. the density of pair of states; one occupied valence state, one empty conduction state). In 3D, this is

but the joint DOS is different for 2D, 1D, and 0D.

Finally we note that in a general way we can express the Fermi golden rule for semiconductors as[13]


Scanning tunneling microscopy Edit

In a scanning tunneling microscope, the Fermi golden rule is used in deriving the tunneling current. It takes the form

where   is the tunneling matrix element.

Quantum optics Edit

When considering energy level transitions between two discrete states, Fermi's golden rule is written as

where   is the density of photon states at a given energy,   is the photon energy, and   is the angular frequency. This alternative expression relies on the fact that there is a continuum of final (photon) states, i.e. the range of allowed photon energies is continuous.[14]

Drexhage experiment Edit

Both the radiation pattern and the total emitted power (which is proportional to the decay rate) of a dipole depend on its distance from a mirror.

Fermi's golden rule predicts that the probability that an excited state will decay depends on the density of states. This can be seen experimentally by measuring the decay rate of a dipole near a mirror: as the presence of the mirror creates regions of higher and lower density of states, the measured decay rate depends on the distance between the mirror and the dipole.[15][16]

See also Edit

References Edit

  1. ^ Bransden, B. H.; Joachain, C. J. (1999). Quantum Mechanics (2nd ed.). Prentice Hall. p. 443. ISBN 978-0582356917.
  2. ^ Dirac, P. A. M. (1 March 1927). "The Quantum Theory of Emission and Absorption of Radiation". Proceedings of the Royal Society A. 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. JSTOR 94746. See equations (24) and (32).
  3. ^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN 978-0226243658. formula VIII.2
  4. ^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN 978-0226243658. formula VIII.19
  5. ^ R Schwitters' UT Notes on Derivation.
  6. ^ It is remarkable in that the rate is constant and not linearly increasing in time, as might be naively expected for transitions with strict conservation of energy enforced. This comes about from interference of oscillatory contributions of transitions to numerous continuum states with only approximate unperturbed energy conservation, see Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) ISBN 0486414620, pp. 150–151.
  7. ^ Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.
  8. ^ Merzbacher, Eugen (1998). "19.7" (PDF). Quantum Mechanics (3rd ed.). Wiley, John & Sons, Inc. ISBN 978-0-471-88702-7.
  9. ^ N. A. Sinitsyn, Q. Niu and A. H. MacDonald (2006). "Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect". Phys. Rev. B. 73 (7): 075318. arXiv:cond-mat/0511310. Bibcode:2006PhRvB..73g5318S. doi:10.1103/PhysRevB.73.075318. S2CID 119476624.
  10. ^ a b c Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (1977). Quantum Mechanics Vol II Chapter XIII Complement D_{XIII}. Wiley. ISBN 978-0471164333.
  11. ^ Bethe, Hans; Salpeter, Edwin (1977). Quantum Mechanics of One- and Two-Electron Atoms. Springer, Boston, MA. ISBN 978-0-306-20022-9.
  12. ^ Yu, Peter Y.; Cardona, Manuel (2010). Fundamentals of Semiconductors - Physics and Materials Properties (4 ed.). Springer. p. 260. doi:10.1007/978-3-642-00710-1. ISBN 978-3-642-00709-5.
  13. ^ Edvinsson, T. (2018). "Optical quantum confinement and photocatalytic properties in two-, one- and zero-dimensional nanostructures". Royal Society Open Science. 5 (9): 180387. Bibcode:2018RSOS....580387E. doi:10.1098/rsos.180387. ISSN 2054-5703. PMC 6170533. PMID 30839677.
  14. ^ Fox, Mark (2006). Quantum Optics: An Introduction. Oxford: Oxford University Press. p. 51. ISBN 9780198566731.
  15. ^ K. H. Drexhage; H. Kuhn; F. P. Schäfer (1968). "Variation of the Fluorescence Decay Time of a Molecule in Front of a Mirror". Berichte der Bunsengesellschaft für physikalische Chemie. 72 (2): 329. doi:10.1002/bbpc.19680720261. S2CID 94677437.
  16. ^ K. H. Drexhage (1970). "Influence of a dielectric interface on fluorescence decay time". Journal of Luminescence. 1: 693–701. Bibcode:1970JLum....1..693D. doi:10.1016/0022-2313(70)90082-7.

External links Edit

  • More information on Fermi's golden rule
  • Derivation of Fermi’s Golden Rule
  • Time-dependent perturbation theory
  • Fermi's golden rule: its derivation and breakdown by an ideal model